A Kaehler manifold of constant holomorphic curvature is called a complex space form. By a Kaehler submanifold we mean a complex submanifold with the induced Kaehler metric. B. Smyth [5] has studied a complete Einstein- Kaehler hypersurface in a complete and simply connected complex space form and classified completely the hypersurface. The local version of this result has been shown to be true by S. S. Chern [1], and partially by T. Takahashi [6] independently.

This paper arises in the attempt to solve the following problem related to the Zariski Problem. Let A be a polynomial ring in three variables over a field, . Suppose there is a subring B of A such that k ⊆ B and there is variable t over B such that B[t] = A. Then is it true that B is a polynomial ring over k?

If H is a (complex) Hilbert space and is a collection of (closed linear) subspaces of H it is easily shown that the set of all (bounded linear) operators acting on H which leave every member of invariant is a weakly closed operator algebra containing the identity operator. This algebra is denoted by Alg . In the study of such algebras it may be supposed [4] that is a subspace lattice i.e. that is closed under the formation of arbitrary intersections and arbitrary (closed linear) spans and contains both the zero subspace (0) and H. The class of such algebras is precisely the class of reflexive algebras [3].

Let be the standard (n + 1) simplex with its standard triangulation. By the Generalized Poincare Conjecture, if n ≧ 5 and is a smooth homotopy w-sphere, then there exists a smooth triangulation , where K is a suitable subdivision of . On the other hand, in [3], N. Kuiper proves the following theorem.

Let A be a commutative real Banach algebra with unit, and MA its maximal ideal space. The existence of the Silov boundary SA for A was established in [5] by resorting to the complexification of A. We give here an intrinsic proof of this result which exhibits the close connection between the absolute values and the real parts of ‘functions’ in A (Theorem 1.3).

We prove that is quasi-primal, then every algebra in HSPhas a pure embedding into a product of finite algebras. For a general theory of varieties for which every can be purely embedded in an equationally compact algebra , and for all notions not explained here, the reader is referred to [38; 6; or 5]. This theorem was known for Boolean algebras simply as a corollary of the Stone representation theorem and the fact that in the variety of Boolean algebras, all embeddings are pure [2].

Let Mn be a Riemannian manifold of dimension n ≧ 2 and class C3, (gtj) the symmetric matrix of the positive definite metric of Mn, and (gij) the inverse matrix of (gtj), and denote by and R = gijRij the operator of covariant differentiation with respect to gij, the Riemann tensor, the Ricci tensor and the scalar curvature of Mn respectively.

Let R be a hereditary noetherian prime ring ((hnp)-ring) with enough invertible ideals. Torsion modules over bounded (hnp)-rings were studied by the author in [10; 11]. All the results proved in [10; 11] also hold for torsion R-modules having no completely faithful submodules. In Section 2, indecomposable injective torsion R-modules which are not completely faithful are studied, and they are shown to have finite periodicities (Theorem (2.8) and Corollary (2.9)). These results are used to determine the structure of quasi-injective and quasi-projective modules over bounded (hnp)-rings (Theorems (2.13), (2.14) and (2.15)).

Let F be a complex analytic subvariety of an open subset of Cn and p ϵ V let be the germs at p of holomorphic, weakly holomorphic, infinitely differentiable, and k times continuously differentiate functions respectively. Spallek [15] has shown that for any p £ V there exists an integer such that , generalizing the result of Malgrange [12] that .In [14], Siu proved Spallek's result from a more sheaf theoretic point of view and showed the minimal integer function is bounded on compact sets. Bloom [7] reproved Malgrange's result by using differential operators on varieties.

We give an example of a centrally symmetric 5-polytope which is linearly stable though its vertices do not form a subset of the vertices of a 5-cube. This example contradicts the “only if” part of the classification theorem on linearly stable poly topes stated by P. McMullen [2]. Moreover the example gives a 5-polytope, the vertices of which form a subset of a 5-cube while its dual does not possess the same property.

Let R be a finite ring with 1 and let R* denote the group of units of R. Let p be a prime number. In this paper we consider the question of whether there exist a, b in R* such that a and b are non- p -th powers whose sum is 1. If such units a, b existing, we say that R is an N (p)-ring. Of course if p does not divide |R*|, the order of R*, then every element in R* is a pth power.

Let G be an (open) domain in the finite complex plane and F a relatively closed proper subset of G. We denote by M(G) the set of functions meromorphic on G and as usual by R(K) (for a compact set K) the set of uniform limits of rational functions without poles on K.

Let ƒ b e a mapping defined on a compact subset K of the finite complex plane C and taking its values on the extended plane C ⋃ ﹛ ∞﹜. For x a metric on the extended plane, we consider the possibility of approximating ƒ x-uniformly on K by rational functions. Since all metrics on C ⋃ ﹛oo ﹜ are equivalent, we shall consider that x is the chordal metric on the Riemann sphere of diameter one resting on a finite plane at the origin.

This article, as the name implies, is a continuation of [9]. In that article the second author investigates finite translation planes containing both affine dations and affine homologies. (See the beginning of Section 2 for definitions.) Such translation planes are called Ei7-planes. In [9] the author restricted himself to translation planes of characteristic p ≧ 5. The main reasons for this were that Ostrom's and Hering's theorem [13;4] on affine dations excluded the case p = 3 and the conclusions were easier to interpret geometrically when p ≧ 5 (as opposed to the case p = 2). Since then Ostrom [17] has settled the case p = 3.

Let A be a C*-algebra and H a Hilbert space of large enough (infinite at least) dimension so that every πƒ, where ƒ is a factor state on A, can be unitarily represented on H. Let Fac (A, H) denote the set of all factor representations of A on H. If π is in Fac (A, H) we call its essential subspace the smallest, closed, vector subspace KoiH such that π (A ) is null on H Θ K. We define Fac∞(A, H) to be the set of elements in Fac (A, H) whose essential subspace is H.

In [1] Ornstein defined a metric on processes which, for processes (P, τ) and (Q, σ) with equal numbers of atoms, measures how closely the motions of P and Q under r and a, respectively, imitate each other. If we think of (P, τ) and (Q, σ) as stationary stochastic processes, and we assume (P, τ) and (Q, σ) are ergodic, then ((P, τ)(Q, σ)) < α says that with probability one a printout from (P, r) can be changed on a set of integers with density less than a to obtain a printout from (Q, σ).

A nonstandard hull of a TVS (locally convex topological vector space) is a standard TVS constructed from a nonstandard model for [3]. If the nonstandard hulls of a TVS are independent of the non-standard model, we say that the TVS has invariant nonstandard hulls. This is (for complete spaces) the property that every finite element is inflnitesimally close to a standard point. We build on the work of Henson and Moore [4], to show that invariance of nonstandard hulls is a self dual property equivalent to bounded sets being precompact, for F and DF spaces, (see Theorem 4.4).

In this paper we consider representations of weakly regular algebras with permutable congruences and a Boolean orthogonality as direct products of orthoprime algebras. Our theorems generalize results of Abian [2; 3] and Speed [24] which characterize direct products of integral domains, and results of Abian [1] and Chacron [7] which characterize direct products of division rings.

In this paper, an induction theorem for rearrangements involving w-tuples in Rn is proved, showing that a certain proposition regarding a pair of w-tuples related by the weak spectral order ≪ is true for any integer n ≧ 2 if and only if it is true for n = 2. This theorem contains as particular cases a well-known theorem of Hardy-Littlewood-Pólya [4, Lemma 2, p. 47], a theorem of Pólya [8], a theorem of Rado [9, pp. 1-2], two theorems of Mirsky [6, Theorem 2, p. 232; 7, Theorem 4, p. 90], a result given in [1, Corollary 2.4, p. 1333] and also [2, Proposition 2.1, p. 439].

Let denote the 2-fold cyclic covering space branched over a link L in S3. We wish to describe an infinite family of prime knots and links in which each member L exhibits two minimal 6-plat representations, where the associated Heegaard splittings of are minimal and inequivalent. Thus each knot or link of that family admits at least two equivalence classes of 6-plat representations which are minimal.